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5: Integration

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    213198

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    • 5.1: Antiderivatives and Indefinite Integrals
      At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function f , how do we find a function with the derivative f and why would we be interested in such a function?
    • 5.2: Approximating Areas
      In this section, we develop techniques to approximate the area between a curve, defined by a function f(x), and the x-axis on a closed interval [a,b]. Like Archimedes, we first approximate the area under the curve using shapes of known area (namely, rectangles). By using smaller and smaller rectangles, we get closer and closer approximations to the area. Taking a limit allows us to calculate the exact area under the curve.
    • 5.3: The Definite Integral
      If f(x) is a function defined on an interval [a,b], the definite integral of f from a to b is given by \[∫^b_af(x)dx=\lim_{n→∞} \sum_{i=1}^nf(x^∗_i)Δx,\] provided the limit exists. If this limit exists, the function f(x) is said to be integrable on [a,b], or is an integrable function. The numbers a and b are called the limits of integration; specifically, a is the lower limit and b is the upper limit. The function f(x) is the integrand, and x is the variable of integration.
    • 5.4: The Fundamental Theorem of Calculus
      The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this is not always easy.
    • 5.5: Substitution
      In this section we examine a technique, called integration by substitution, to help us find antiderivatives. Specifically, this method helps us find antiderivatives when the integrand is the result of a chain-rule derivative.
    • 5.6: The Net Change Theorem
      The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero. The area under an even function over a symmetric interval can be calculated by doubling the area over the positive x-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.
    • 5.7: Integration of Hyperbolic Functions
      We were introduced to hyperbolic functions in Introduction to Functions and Graphs, along with some of their basic properties. In this section, we look at differentiation and integration formulas for the hyperbolic functions and their inverses.
    • 5.R: Chapter 5 Review Exercises
      This page covers calculus topics, focusing on antiderivatives and integrals. It includes finding antiderivatives, true/false statements on integral properties, Riemann sums, and evaluations of definite integrals. The relationship between derivatives and integrals is discussed, concluding with an analysis of historical average costs of RAM related to a bullet's velocity function and its average velocity over a time interval.

    This page titled 5: Integration is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.